复数与四元数的初等关联定理

An elementary proof of a matrix tree theorem for directed graphs

We present an elementary proof of a generalization of Kirchoff's matrix tree theorem to directed, weighted graphs. The proof is based on a specific factorization of the Laplacian matrices associated to the graphs, which only involves the two incidence matrices that capture the graph's topology. We also point out how this result can be used to calculate principal eigenvectors of the Laplacian matrices.

Incidences between points and lines in three dimensions

We give a fairly elementary and simple proof that shows that the number of incidences between $m$ points and $n$ lines in ${\mathbb R}^3$, so that no plane contains more than $s$ lines, is $$ O\left(m^{1/2}n^{3/4}+ m^{2/3}n^{1/3}s^{1/3} + m + n\right) $$ (in the precise statement, the constant of proportionality of the first and third terms depends, in a rather weak manner, on the relation between $m$ and $n$). This bound, originally obtained by Guth and Katz~\cite{GK2} as a major step in their solution of Erd{\H o}s's distinct distances problem, is also a major new result in incidence geometr…

关于投影平面室的克罗内克图着色数的注记

A note on the chromatic number of Kneser graphs on chambers of projective planes and incidence-free sets

本文通过简单证明展示了在投影平面的 incidence 图中，X 和 Y 的补集之间存在完美匹配，从而将投影平面室的克罗内克图着色数问题与 incidence 图的 incidence-free 数问题联系起来。

点-超平面关联几何与对数秩猜想

Point-hyperplane incidence geometry and the log-rank conjecture

本文从点-超平面关联几何角度提出并证明一个新猜想与对数秩猜想等价，同时改进无结构构型中完全二部子图密度下界与上界构造，为理解改进Lovett上界的困难提供新框架。

更多具有许多富线的点集

More pointsets with many rich lines

本文提出了Szemerédi-Trotter定理的新锐利构造，推广了先前工作，并用纯关联几何论证取代数论论证，获得更简单的证明和更一般的结果。

关于codegree Turán密度任意接近零的线性$k$图

On linear $k$-graphs with codegree Turán density arbitrarily close to zero

本文构造了任意$k$上的线性$k$图，其codegree Turán密度可以任意小，解决了$k=3$时Ding等人提出的问题。

Sylvester-Gallai定理、着色与代数

The Sylvester-Gallai Theorem, colourings and algebra

我们的出发点是以下简单的共同推广西尔维斯特-加莱定理和莫茨金-拉宾定理：设S是一个有限集的点在平面上，与每个点的颜色红色或蓝色或与两种颜色。假设对于S中共享一种颜色的任何两个不同的点A，B，存在S中另一种颜色的第三个点C，与A和B共线。那么S中的所有点都是共线的。我们定义一个色几何是一个简单的拟阵，其中每个点都是红色或蓝色或两种颜色，使得对于S中共享一种颜色的任何两个不同的点A，B，S中有第三个点C，另一种颜色，与A和B共线。这是真有限线性空间和真双色有限线性空间的一个普通推广，两者都有许多已知的推广性质。一个这样的性质是凯利的复Sylvester-Gallai定理。我们还考虑了色几何在Deskuesian射影空间中的嵌入。本文证明了四元数上的射影空间中三维色几何的点数的下界为51。最后，我们提出了一个初等的方法来推论的不等式Hirzebruch凯利在他的证明复杂的Sylvester-Gallai定理。